Irregular Hodge theory.

*(English. French summary)*Zbl 1422.14003Let \(X\) be a complex manifold, and let \(\mathcal{D}_X\) and \(\Theta_X\) be the sheaf of holomorphic differential operators on \(X\) and the tangent sheaf of \(X\), respectively. We denote by \(\mathcal{R}^{\operatorname{int}}_{\mathbb{C}_z\times X}\) the sheaf of Rees rings on \(X\) with integrable structure, that is, the subsheaf of non-commutative algebras of \(\mathcal{D}_{\mathbb{C}_z\times X}\) generated by \(zp^*\Theta_X\) and \(z^2\partial_z\), where \(p:\mathbb{C}_z\times X \rightarrow X\) is the projection. (If \((x_1,\ldots,x_n)\) are local coordinates on \(X\), then \(\mathcal{R}^{\operatorname{int}}_{\mathbb{C}_z\times X}\) is locally given by \(\mathcal{O}_{\mathbb{C}_z\times X}\langle z^2\partial_z, z\partial_{x_1},\ldots,z\partial_{x_n}\rangle\).) Let us denote by \(\operatorname{MHM}(X)\) the abelian category of mixed Hodge modules on \(X\).

We will also write \(\operatorname{MTM}^{\operatorname{int}}(X)\) to denote the abelian category of integrable mixed twistor \(\mathcal{D}\)-modules on \(X\) (see [T. Mochizuki, Lecture Notes in Mathematics 2125. Cham: Springer (2015; Zbl 1356.32002)]). The objects of this category are triples consisting of two \(\mathcal{R}_{\mathbb{C}_z\times X}^{\operatorname{int}}\)-modules on \(X\), together with a pairing relating certain restrictions of them.

The book under review introduces a new category, \(\text{IrrMHM}(X)\), which is a certain subcategory of \(\operatorname{MTM}^{\text{int}}(X)\) (actually of a variant, called \({_\iota\!}\operatorname{MTM}^{\text{int}}(X)\), which is shown to be equivalent to \(\operatorname{MTM}^{\text{int}}(X)\)) consisting of objects \(\widehat{\mathcal{M}}\) that satisfy certain properties. The first of them is that the object \({^\theta}\widehat{\mathcal{M}}\) obtained from \(\widehat{\mathcal{M}}\) by substituting \(z\theta\) for \(z\) is still an object of \({_\iota\!}\operatorname{MTM}^{\text{int}}({}^\theta\! X)\), where \({}^\theta\! X=\mathbb{C}_{\theta}^*\times X\). That is a remarkable assumption, having its origin in [C. Hertling and Ch. Sevenheck, J. Reine Angew. Math. 609, 23–80 (2007; Zbl 1136.32011)], but we have to impose a further property, namely that such rescaled objects must have a certain tame behaviour when \(\theta\) goes to infinity (or at the origin of \(\tau=1/\theta\), in other words). This two conditions and a \(z\)-grading property appearing after identifying \(\tau\) with \(z\) are, essentially, what we ask for a integrable mixed twistor \(\mathcal{D}\)-module on \(X\) to become an irregular mixed Hodge module.

Although the construction of the new category may seem rather involved, its main feature is that the \(\mathcal{D}_X\)-module \(\mathcal{M}\) associated to an object \(\widehat{\mathcal{M}}\) in \(\operatorname{IrrMHM}(X)\) carries a good filtration \(F_\bullet^{\text{irr}} \mathcal{M}\), indexed by \(\mathbb{R}\), called the irregular Hodge filtration, which in turn behaves well with respect to several functorial operations. Namely, the category is preserved by direct image by a projective morphism, inverse image by a smooth morphism, tensor product, duality, localization along a divisor and, unlike mixed Hodge modules, exponential twist, which allows working with the Fourier-Laplace transformation.

Moreover, one of the main results is that any mixed twistor \(\mathcal{D}\)-module on \(X\) obtained from a mixed Hodge module by applying any combination of the above functors is an object of \(\operatorname{IrrMHM}(X)\). Besides, mixed Hodge modules can be endowed with a canonical structure of irregular mixed Hodge module such that the ordinary Hodge filtration is also the irregular one.

Another important application of this theory is the proof that any rigid, irreducible, locally formally unitary \(\mathcal{D}\)-module on \(\mathbb{P}^1\) can be endowed with a unique structure of irregular mixed Hodge module, carrying thus a canonical irregular Hodge filtration. It can be seen as an extension of the case of regular singularities of [C. Simpson, J. Am. Math. Soc. 3, No. 3, 713–770 (1990; Zbl 0713.58012)] to the general irregular setting.

The first chapter of the book deals with the construction of the category \({_\iota\!}\operatorname{MTM}^{\text{int}}(X)\), which is shown to be functorially equivalent to \(\operatorname{MTM}^{\text{int}}(X)\). The difference between both categories lies in where the restrictions of the \(\mathcal{R}_{\mathbb{C}_z\times X}^{\text{int}}\)-modules are taken when considering the pairing: in \(\mathbf{S}^1\times X\) in the latter, or in \(\mathbb{C}_z^*\times X\) in the former, so that the rescaling operation can make sense.

In the second chapter, the author defines the rescaling operation, shows its behaviour under all relevant functors and states the conditions referred to above on his way to the definition of irregular mixed Hodge modules. Some explanatory examples are included, as well as the proof of the existence and uniqueness of the irregular mixed Hodge module structure for rigid, irreducible \(\mathcal{D}_{\mathbb{P}^1}\)-modules, based on the Katz-Arinkin algorithm.

The third and last chapter, written in collaboration with Jeng-Daw Yu, is devoted to irregular mixed Hodge structures. There, a Tannakian formalism and some compatibilities with the existent categories of exponential mixed Hodge structures and mixed twistor structures are given. The authors also give an alternative proof of a formula giving the irregular Hodge numbers of a purely irregular hypergeometric \(\mathcal{D}\)-module, part of an original result of the reviewer and Ch. Sevenheck [J. Inst. Math. Jussieu First View, 1–42 (2019; doi:10.1017/S1474748019000288)], as well as a Künneth formula and a Thom-Sebastiani theorem for the irregular Hodge filtration.

The book is well written and offers a very interesting reading for any expert in fields such as \(\mathcal{D}\)-modules, Hodge theory or motives, among others.

We will also write \(\operatorname{MTM}^{\operatorname{int}}(X)\) to denote the abelian category of integrable mixed twistor \(\mathcal{D}\)-modules on \(X\) (see [T. Mochizuki, Lecture Notes in Mathematics 2125. Cham: Springer (2015; Zbl 1356.32002)]). The objects of this category are triples consisting of two \(\mathcal{R}_{\mathbb{C}_z\times X}^{\operatorname{int}}\)-modules on \(X\), together with a pairing relating certain restrictions of them.

The book under review introduces a new category, \(\text{IrrMHM}(X)\), which is a certain subcategory of \(\operatorname{MTM}^{\text{int}}(X)\) (actually of a variant, called \({_\iota\!}\operatorname{MTM}^{\text{int}}(X)\), which is shown to be equivalent to \(\operatorname{MTM}^{\text{int}}(X)\)) consisting of objects \(\widehat{\mathcal{M}}\) that satisfy certain properties. The first of them is that the object \({^\theta}\widehat{\mathcal{M}}\) obtained from \(\widehat{\mathcal{M}}\) by substituting \(z\theta\) for \(z\) is still an object of \({_\iota\!}\operatorname{MTM}^{\text{int}}({}^\theta\! X)\), where \({}^\theta\! X=\mathbb{C}_{\theta}^*\times X\). That is a remarkable assumption, having its origin in [C. Hertling and Ch. Sevenheck, J. Reine Angew. Math. 609, 23–80 (2007; Zbl 1136.32011)], but we have to impose a further property, namely that such rescaled objects must have a certain tame behaviour when \(\theta\) goes to infinity (or at the origin of \(\tau=1/\theta\), in other words). This two conditions and a \(z\)-grading property appearing after identifying \(\tau\) with \(z\) are, essentially, what we ask for a integrable mixed twistor \(\mathcal{D}\)-module on \(X\) to become an irregular mixed Hodge module.

Although the construction of the new category may seem rather involved, its main feature is that the \(\mathcal{D}_X\)-module \(\mathcal{M}\) associated to an object \(\widehat{\mathcal{M}}\) in \(\operatorname{IrrMHM}(X)\) carries a good filtration \(F_\bullet^{\text{irr}} \mathcal{M}\), indexed by \(\mathbb{R}\), called the irregular Hodge filtration, which in turn behaves well with respect to several functorial operations. Namely, the category is preserved by direct image by a projective morphism, inverse image by a smooth morphism, tensor product, duality, localization along a divisor and, unlike mixed Hodge modules, exponential twist, which allows working with the Fourier-Laplace transformation.

Moreover, one of the main results is that any mixed twistor \(\mathcal{D}\)-module on \(X\) obtained from a mixed Hodge module by applying any combination of the above functors is an object of \(\operatorname{IrrMHM}(X)\). Besides, mixed Hodge modules can be endowed with a canonical structure of irregular mixed Hodge module such that the ordinary Hodge filtration is also the irregular one.

Another important application of this theory is the proof that any rigid, irreducible, locally formally unitary \(\mathcal{D}\)-module on \(\mathbb{P}^1\) can be endowed with a unique structure of irregular mixed Hodge module, carrying thus a canonical irregular Hodge filtration. It can be seen as an extension of the case of regular singularities of [C. Simpson, J. Am. Math. Soc. 3, No. 3, 713–770 (1990; Zbl 0713.58012)] to the general irregular setting.

The first chapter of the book deals with the construction of the category \({_\iota\!}\operatorname{MTM}^{\text{int}}(X)\), which is shown to be functorially equivalent to \(\operatorname{MTM}^{\text{int}}(X)\). The difference between both categories lies in where the restrictions of the \(\mathcal{R}_{\mathbb{C}_z\times X}^{\text{int}}\)-modules are taken when considering the pairing: in \(\mathbf{S}^1\times X\) in the latter, or in \(\mathbb{C}_z^*\times X\) in the former, so that the rescaling operation can make sense.

In the second chapter, the author defines the rescaling operation, shows its behaviour under all relevant functors and states the conditions referred to above on his way to the definition of irregular mixed Hodge modules. Some explanatory examples are included, as well as the proof of the existence and uniqueness of the irregular mixed Hodge module structure for rigid, irreducible \(\mathcal{D}_{\mathbb{P}^1}\)-modules, based on the Katz-Arinkin algorithm.

The third and last chapter, written in collaboration with Jeng-Daw Yu, is devoted to irregular mixed Hodge structures. There, a Tannakian formalism and some compatibilities with the existent categories of exponential mixed Hodge structures and mixed twistor structures are given. The authors also give an alternative proof of a formula giving the irregular Hodge numbers of a purely irregular hypergeometric \(\mathcal{D}\)-module, part of an original result of the reviewer and Ch. Sevenheck [J. Inst. Math. Jussieu First View, 1–42 (2019; doi:10.1017/S1474748019000288)], as well as a Künneth formula and a Thom-Sebastiani theorem for the irregular Hodge filtration.

The book is well written and offers a very interesting reading for any expert in fields such as \(\mathcal{D}\)-modules, Hodge theory or motives, among others.

Reviewer: Alberto Castaño Domínguez (Sevilla)

##### MSC:

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |

32C38 | Sheaves of differential operators and their modules, \(D\)-modules |

14F40 | de Rham cohomology and algebraic geometry |

32S35 | Mixed Hodge theory of singular varieties (complex-analytic aspects) |

32S40 | Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) |

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |